Mobility in Symmetry-Regular Bar-and-Joint Frameworks
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In a symmetry-regular bar-and-joint framework of given point-group symmetry, all bars and joints occupy general positions with respect to the symmetry elements. The symmetry-extended form of Maxwell’s Rule is applied to this simplest type of framework and is used to derive counts within irreducible representations for infinitesimal mechanisms and states of self stress. In particular, conditions are given for symmetry-regular frameworks to have at least one infinitesimal mechanism (respectively, state of self stress) within each irreducible representation of the point group of the framework. Similar conditions are found for symmetry-regular body-and-joint frameworks.
KeywordsInfinitesimal rigidity Mechanisms Frameworks Symmetry
Subject Classifications52C25 70B15 20C35
This work was begun during the workshop on Rigidity and Symmetry, held at the Fields Institute, Toronto, October 17–21, 2011, and gained from discussions there with Prof. Stephen Power (Lancaster). PWF thanks the Fields Institute for financial support of his visit.
- 1.Altmann, S.L., Herzig, P.: Point-Group Theory Tables. Clarendon Press, Oxford (1994)Google Scholar
- 2.Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory. Oxford University Press, London (1970)Google Scholar
- 3.Bishop, D.M.: Group Theory and Chemistry. Clarendon Press, Oxford (1973)Google Scholar
- 9.Fowler, P.W., Guest, S.D.: A symmetry analysis of mechanisms in rotating rings of tetrahedra. Proc. R. Soc.: Math. Phys. Eng. Sci. 461, 1829–1846 (2005)Google Scholar
- 22.Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Magazine. 27, 294–299 (1864)Google Scholar